The present invention relates to a filler arrangement comprising a first filter having an amplitude characteristic that meets a redefined amplitude specification and having a phase characteristic that generally is a non-linear function of frequency and a second filter, cascade coupled to the first filter, the second filter having a phase characteristic that is, up to a linear function of frequency, substantially opposite to the phase characteristic of the first filter a and a method to develop such a filter arrangement by designing a first filter so that its amplitude characteristic meets a predefined amplitude specification implementing the first filter, determining a phase characteristic of the first filter, the phase characteristic being a non-linear function of frequency, implementing a second filter so that its phase characteristic is up to a linear function of frequency, substantially opposite to the phase characteristic of the first filter and cascade coupling the first filter and second filter.
Filter arrangements with linear phase characteristics are well-known in the art, and are typically constituted of a first filter with predefined amplitude characteristic and a cascade coupled second filter that is an all-pass filter which has approximately the opposite phase characteristic of the first filter (up to a linear function of frequency). The first filter may be an IIR (Infinite Impulse Response) filter that meets the predefined amplitude characteristics with a relatively low filter order, e.g. a third or fourth order filter, but whose phase characteristic is not a linear function of frequency. Such a filter causes non-linear phase distortions which are not acceptable in several applications. This is for instance so in a multi-carrier transmission system, such as an ADSL (Asymmetric Digital Subscriber Line) system based on DMT (Discrete Multi Tone) modulation, if peak/average power ratio (PAR) reduction techniques are applied because the non-linear phase distortions caused by the filter destroy most of the obtained PAR reduction. The phase distortion introduced by the first filter, designed to meet a given amplitude specification, is thus compensated for by a second filter. In the known filter arrangements with linear phase characteristic, this second filter is a causal digital all-pass filter (that is a digital all-pass filter whose current output sample depends on the current and previous input samples and on previous output samples) whose phase characteristic is designed to oppose, up to a linear function of frequency, the phase characteristic of the first filter. The phase of the cascade connection of the first filter and the causal all-pass filter is the sum of the phases of both filters and consequently is an approximately linear function of frequency, whereas the amplitude characteristic of the cascade connection of both filters is equal to the amplitude characteristic of the first filter. However, when the phase distortion of the first filter is severe, one typically needs a causal all-pass filter of relatively high order, e.g. a seventh or eighth order IIR all-pass filter, to achieve a good phase equalisation. Such high order all-pass filters are difficult to implement.